Absence of Critical Points of Solutions to the Helmholtz Equation in 3D

Abstract

The focus of this paper is to show the absence of critical points for the solutions to the Helmholtz equation in a bounded domain $Ω\subset\mathbb{R}^{3}$, given by \[ \left\{ \begin{array}{l} -\rm{div}(a\,\nabla u_ω^{g})-ωqu_ω^{g}=0\quad\text{in $Ω$,}\\ u_ω^{g}=g\quad\text{on $\partialΩ$.} \end{array}\right. \] We prove that for an admissible $g$ there exists a finite set of frequencies $K$ in a given interval and an open cover $\overlineΩ=\cup_{ω\in K}Ω_ω$ such that $|\nabla u_ω^{g}(x)|>0$ for every $ω\in K$ and $x\inΩ_ω$. The set $K$ is explicitly constructed. If the spectrum of the above problem is simple, which is true for a generic domain $Ω$, the admissibility condition on $g$ is a generic property.